Experimental study of the Doppler shift generated by a vibrating scatterer Régis Wunenburger, a) Nicolás Mujica, b) and Stéphan Fauve Laboratoire de Physique Statistique, Ecole Normale Supérieure, CNRS UMR 8550, 24 rue Lhomond, 75231 Paris Cedex 05, France Received 9 May 2003; revised 28 October 2003; accepted 31 October 2003 We report an experimental study of the backscattering of a sound wave of frequency f by a surface vibrating harmonically at frequency F (F f ) and amplitude A in the regime where the Doppler effect overcomes bulk nonlinear effects. When the duration t 0 of the analyzed time series of the scattered wave is small compared to the vibration period, the power spectrum of the backscattered wave is proportional to the probability density function of the scatterer velocity, which presents two peaks shifted from f by roughly 2 fa /c ( 2 F). On the contrary, when t 0 F 1, sidebands at frequencies f nf n integer appear in the power spectrum, which are due to the phase modulation of the backscattered wave induced by its reflection on a moving boundary. We use the backscattered power spectrum to validate the phase modulation theory of the Doppler effect in the latter case for 2kA 1 and 2kA 1 (k 2 f /c, where c is the wave velocity and we test the validity of an acoustic nonintrusive estimator of A as a function of power spectrum bandwidth and of A itself. 2004 Acoustical Society of America. DOI: 10.1121/1.1635414 PACS numbers: 43.20.Fn, 43.28.Py, 43.25.Lj MFH Pages: 507 514 I. INTRODUCTION a Permanent address: Center de Physique Moléculaire Optique et Hertzienne, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France. Electronic mail: r.wunenburger@cpmoh.u-bordeaux1.fr b Permanent address: Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Av. Blanco Encalada 2008, Santiago, Chile. Present address: Nonlinear Dynamics Laboratory, Institute for Research in Electronics and Applied Physics, Bldg 223, University of Maryland, College Park, MD 20742. Velocity measurement using the Doppler shift of a wave reflected from a moving object is a widely used technique both with electromagnetic and acoustic waves. For objects moving at constant velocity, it is well known that the Doppler shift, calculated by means of coordinate transformation, is proportional to the velocity. The problem is more difficult when the motion is time dependent. In the case of a periodically oscillating object the problem was first carefully studied for electromagnetic waves. 1 The spectrum of the scattered wave is also modified due to the Doppler effect, which can be understood either as a nonlinear boundary condition imposed by the moving object 1,2 or as caused from the inhomogeneity in time of the moving medium that supports the wave propagation. 3 More precisely, the spectrum of the wave at frequency f scattered by a sinusoidally oscillating surface at frequency F is similar to that of a phase modulation process, 1 i.e., sidebands at frequencies f nf n integer appear in the spectrum of the scattered wave. In the case of acoustic scattering, the situation is more complex, as the oscillating scatterer also emits a sound wave at frequency F which interacts with the scattered wave due to the nonlinear character of the equations of sound propagation. 4,5 Rogers remarked 6 that the bulk nonlinear wave mixing produces the same sideband peaks in the spectrum of the backscattered sound wave as the Doppler effect. Due to the lack of any decisive experiment, the criteria allowing us to discriminate between the two effects have been the subject of an intense debate, 3,6 11 and it has even been claimed that the contribution of the Doppler effect might be undetectable. 6,8 11 In a previous letter, 12 we showed that there exists a wide parameter range in which the Doppler shift gives the dominant contribution to the spectrum of the scattered wave and studied the cross-over to the bulk dominated nonlinear regime. In this article we focus on the experimental situation where the Doppler effect induced by a vibrating scatterer overcomes bulk nonlinearities. In particular we study the backscattering of a high frequency f sound wave by a plane scatterer oscillating at low frequency F f. We study the characteristic features of the Doppler effect in both the static and quasi-static regimes and we validate the phase modulation theory of the Doppler effect in the quasistatic regime for 2kA 1 and 2kA 1. Many studies have been devoted to vibration measurements using ultrasonic techniques. 13 18 In particular, Huang et al. proposed an acoustic nonintrusive estimator of the scatterer oscillation amplitude based on the phase modulation of the backscattered wave. 15 To our knowledge, this estimator has not been validated experimentally. By comparing the scatterer oscillation amplitude obtained with both the acoustic nonintrusive estimator and vibration measurements performed with an accelerometer, we verify the accuracy of Huang et al. s amplitude estimator. This paper is organized as follows: in Sec. II we present the conditions for observing the static and quasi-static Doppler effect, and we recall some predictions concerning both surface Doppler and bulk nonlinear effects. This helps us to justify our choice of the experimental configuration, which is presented in Sec. III. In Sec. IV we present the main features of the static and quasi-static Doppler effect. In Sec. V we finally test the validity of a nonintrusive estimator of the J. Acoust. Soc. Am. 115 (2), February 2004 0001-4966/2004/115(2)/507/8/$20.00 2004 Acoustical Society of America 507
scatterer oscillation amplitude. Conclusions are given in Sec. VI. p D exp i t 2kA sin t L c, 4 II. THEORETICAL CONSIDERATIONS A. Static versus quasi-static Doppler effect The usual picture of the Doppler effect is the constant frequency shift of an incident or emitted wave by an object moving at constant velocity V, which we will call the static Doppler effect. In the case of backscattering, the frequency shift f encountered by the scattered wave of frequency f is 2 fv/c. From an experimental point of view, when the velocity of the scatterer varies, the latter approach remains valid when the timescale of velocity variations is much larger than the duration t 0 of the scattered wave time series which is analyzed. For a periodic motion of frequency F, this implies t 0 F 1. In this case, the statistical distribution of the successively measured Doppler shifts is proportional to the probability density function PDF of the object velocity. In practice, since f and F are not commensurate, the computed power spectrum of the backscattered wave is the average of power spectra of many successive signal time series of duration t 0 (t 0 f 1 ) measured at random phase of the scatterer motion. Thus, this time-averaged power spectrum is expected to be proportional to the PDF of the scatterer velocity. When t 0 is large enough so that the scatterer velocity varies during the acquisition, an analysis in terms of modulation of the time of flight of the scattered wave shows that the wave is phase modulated. To show this, we will consider the unidimensional situation where a plane progressive monochromatic sound wave propagating at velocity c in a quiescent medium is backscattered by a plane scatterer. This object has an infinite acoustic impedance normal total reflection and oscillates sinusoidally around its mean position according to the trajectory x S (t) A sin t. We assume that the wave is emitted at time t (t) by a transducer located at a distance L from the scatterer. The backscattered wave is then detected by the same transducer at time t such that t 2 c L x S t t 2. 1 This modulation of the time of flight of the wave induces a phase modulation of the backscattered wave detected at a distance L from the scatterer, such that p D exp i t t, 2 where the superscript D denotes the Doppler contribution to the backscattered wave. If A L, we have (t) 2L/c. In addition, if the velocity of the plate is small compared to the sound velocity c, M A /c 1, substitution of (t) 2L/c in the argument of the vibration amplitude x S in 1 is justified and leads to an explicit approximate expression for (t). This gives a phase modulation of the form t 2 c L x S t L c. 3 This is what is called the quasi-static approximation of the Doppler effect. The detected wave then rewrites where k /c is the wave number of the high frequency incident acoustic wave. The generation of sideband peaks at pulsation n n integer in the spectrum of the backscattered wave is evidenced when transforming the latter expression to p D exp i t n J n 2kA exp in t L 5 c, where J n is the Bessel function of nth order. The sideband peaks at frequency f nf (F /2 ) have their amplitude proportional to J n (2kA). For 2kA 1, J n (2kA) (ka) n /n!, and the Doppler effect is considered as weak, i.e., the energy of the peak at frequency f in the spectrum of the backscattered wave is almost equal to the energy of the incident wave, and the leading sideband peaks are at frequency f F. Note that the condition M 1 implies two possible situations: the first is F f, thus Eq. 5 is valid for any value of 2kA such that 2kA f/f; the second is F f, thus A /c A /c 1, and therefore Eq. 5 is restricted to ka 1. Finally, the general case of oblique incidence has been widely studied theoretically. 1,3,7,19,20 If we define as the angle between the incident wave and the normal of the surface, the argument of the Bessel functions of the scattered wave Eq. 5 must be replaced by 2kA k z k nz A, 6 where k z k cos, k /c, k nz n 2 c k 2 x, 7 and k x k sin. The argument (k z k nz )A depends then on n and. IfF f and if we restrict to the first sidebands, we can approximate ( n )/c /c in 7. Thus, the argument of the Bessel functions results, k z k nz A 2kA cos, 8 and, for small angles, the multiplicative correction term is almost unity, cos 1 2 /2. Thus, from an experimental point of view, a simple configuration to study is 1 and F f ; in this case, Eq. 5 holds for any value of 2kA such that 2kA f /F. B. Surface quasi-static Doppler effect versus bulk nonlinear effects Due to the intrinsic nonlinear character of the conservation equations and to the nonlinear dependence of pressure fluctuations against density fluctuations as a consequence of the equation of state of the fluid, sound propagation is nonlinear. 4,5 Thus, two collinear waves of frequencies f 1 and f 2 may interact and generate waves whose frequencies are linear combinations of f 1 and f 2, and whose amplitudes increase with the distance of interaction L. 5 Considering our experiment, the nonlinear interaction of the low frequency wave p emitted by the vibrating scatterer and the high frequency backscattered wave p leads to the generation of 508 J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer
sideband peaks in the spectrum of the detected wave. In the case of weak nonlinear interaction, the amplitude of the first sideband is to leading order proportional to NL p 2 c p p 3 L, where B/2A 1 is the usual nonlinear parameter of the medium 5,21 here A should not be confused with the vibration amplitude. For gases, 1 /2, with c p /c v the specific heat ratio. Therefore, when an acoustic wave is scattered by a vibrating surface, both the surface Doppler effect and bulk non-linearities contribute to the generation of combination frequencies. However, it has been argued that the Doppler effect is practically always dominated by bulk nonlinearities. 6,8 11 In the case of plane waves, Piquette and Van Buren 10 and later Bou Matar et al. 18 predicted that when the dimensionless parameter Y p D NL p 2 c2 A Lp 9 10 is large resp. small compared to unity, the Doppler effect resp. bulk nonlinear effect is dominant. Note that in addition to the plane wave assumption, such that p cv, these studies use the approximation that the amplitude of the low frequency velocity field is given by the scatterer velocity, i.e., v A, obtaining Y / L, where c/f. Completing the experimental demonstration by Bou Matar 18 of the existence of two asymptotic regimes, a Doppler dominant regime and a bulk nonlinearity dominant one, we showed in our previous experimental study 12 that the criterium Y 1, expressed in its more general form Eq. 10, is quantitatively correct. 22 In this article, we focus on the Doppler effect, and therefore we choose an experimental configuration where the condition Y 1, as defined by Eq. 10, is verified. III. EXPERIMENTAL SETUP Figure 1 displays the experimental setup. The vibrating scatterer consists of a square flat piston made of PMMA, 17 17 cm and 10 mm thick, located at x 0 say. It is driven sinusoidally by an electromechanical vibration exciter of Bruel & Kjaer BK 4808 type, at a frequency F 14 Hz and amplitude 10 6 m A 3.5 10 3 m. An air coupled transducer ITC 9073, d 12 mm in diameter, located at a distance L 28 cm, generates a wave at frequency f 225 khz, incident on the vibrating plane with a small angle 5. It is driven by the source of a high frequency spectrum analyzer Agilent 3589 amplified by a NF Electronic Instruments 4005 power amplifier. The incident wave on the piston is in the far field, as d 2 /4 23.5 mm L. The backscattered wave is detected by another ITC 9073 transducer, also located at a distance L, and oriented with an angle 5. The displacement of the piston is controlled by a Wavetek 395 function generator. This sinusoidal electric signal is amplified by a BK 2712 power amplifier, and the displacement is of the form x S (t) A sin( t). We note that the vibration harmonics FIG. 1. Sketch of the experimental apparatus. VE electromechanical vibration exciter, P 17 17 1 cm PMMA piston, A piezoelectric accelerometer, T1-T2 air coupled transducers, AV anti-vibrations supports, and F foam anti-vibration layer. In the experiments described here, i r 5 and L 280 mm. The piston displacement is x S (t) A sin( t). have an energy at least 100 times smaller that the fundamental frequency vibration. The acceleration A 2 sin( t) of the scatterer is measured using a BK 4393V piezoelectric accelerometer and a BK 2635 charge amplifier. The acceleration signal is processed using both an Agilent 35670A low frequency spectrum analyzer and a Stanford Research System 830 Lock-In amplifier, in order to get the displacement of the scatterer at F the difference between both measurements is smaller than 0.5%. The power spectrum of the backscattered wave is computed with the high frequency Agilent 3589A spectrum analyzer. The experiment is controlled by a Power PC Mac computer and the data are transferred to this computer via a general purpose interface bus GPIB board. J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer 509
Concerning the power spectrum measurements, an important parameter is t 0 F, where t 0 is the time duration of the analyzed time series, and t 0 1 is thus the spectral resolution SR. In practice, it is the frequency span, f sp, of the Agilent spectrum analyzer which is varied for a fixed number of points, N 401. Thus, t 0 1 f sp /(N 1), and the frequency span takes the values 5 khz, 2.5 khz, 1.25 khz, 625 Hz, and 312.5 Hz, giving t 0 1 12.5, 6.25, 3.125, 1.5625, and 0.78125 Hz. Finally, in our experiment 5 is finite but small. As discussed in Sec. II, if F f and if we restrict our analysis to the first sidebands, the argument of the Bessel functions appearing in Eq. 5 must be replaced by 2kA cos see Eq. 8. This is indeed the case in our experiments, since F O(10 Hz), f O(225 khz), and the number of analyzed sidebands is either 5 or 11. For 5, the correct argument of the Bessel functions is 2kA cos 1.992kA, which represents a 0.4% error with respect to 2kA. This error is small compared to the errors related to the measurements of A and the error on k given by the approximation c 345 5 m/s at 25 C, and it is then considered as negligible. Therefore, we simply use the parameter 2kA in the analysis of our experimental results. IV. MAIN CHARACTERISTIC FEATURES OF THE DOPPLER EFFECT A. Static Doppler effect As explained above, when t 1 0 F the time averaged backscattered wave power spectrum is expected to be proportional to the PDF of the scatterer velocity. In addition, a static Doppler shift can be detected only if it exceeds the spectral resolution. This implies an additional condition, 2 fa /c t 1 0, i.e., 2kA 1. In our experimental setup both conditions are simultaneously verified for F 5 Hz and A 1.7 mm. In addition Y 1, and we are then in the Doppler dominated regime. Figure 2 a displays the backscattered wave power spectrum density PSD obtained for a low spectral resolution SR of t 1 0 12.5 Hz, such that t 1 0 2.5F, and a scatterer vibration amplitude A verifying 2kA 32.6. Thus, the conditions for observing a static Doppler shift are approximately verified. We observe that the spectrum displays two maxima, and we define f as the frequency shift of these maxima; in this case f 150 Hz. These maxima can be identified with the two maxima of the scatterer velocity PDF occurring at A. Its theoretical expression FIG. 2. a Backscattered wave power spectrum density PSD for F 5 Hz and t 0 1 12.5 Hz. Frequencies are shifted by f 225 khz. b Experimental and theoretical solid line PDFs of the normalized scatterer velocity. the expected static Doppler frequency shift 2 fa /c. The solid line represents the expected correlation between these two quantities in the static Doppler effect regime. The systematic underestimation of f is probably due to the fact that the scatterer velocity is not constant during the time of signal acquisition. Nevertheless, there is a satisfactory correlation between both quantities. Figure 4 displays the backscattered wave power spectra 1 PDF ẋ S /A sin arccos ẋ S /A 11 is presented in Fig. 2 b together with the experimental PDF of the actual scatterer velocity normalized by the value A 0.125 m/s, obtained with the acceleration measurement done with the lock-in amplifier. Note that in Fig. 2 a two pairs of extra symmetric peaks appear at roughly 2 f and 3 f, but they are of much lower intensity. They probably correspond to the Doppler shifts of the waves which undergo multiple reflections on the scatterer. Figure 3 displays the variations of f as a function of FIG. 3. Maximal frequency shift measured from the backscattered wave power spectrum f as a function of 2A f /c which is the expected static Doppler shift obtained from acceleration measurements. Error bars correspond to the frequency resolution t 0 1 12.5 Hz and F 5 Hz. Solid line represents f 2A f. 510 J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer
FIG. 4. Backscattered wave power spectra for F 5 Hz and t 0 1 12.5 Hz 1, 6.25 Hz 2, 3.125 Hz 3, 1.5625 Hz 4, and 0.78125 Hz 5. Frequencies were shifted by f 225 khz. Curves 3 5 have been shifted down by 5, 15, and 40 db, respectively, for a better display. obtained for various spectral resolutions ranging from t 0 1 12.5 to 0.78125 Hz. For a larger SR, t 0 1 6.25 Hz F, the condition of static Doppler effect is not verified, and curve 2 of Fig. 4 displays several peaks between the two maxima of curve 1. The number of peaks increases with the spectral resolution curves 3 and 4 until the frequency difference between two peaks drops to F 5 Hz curve 5. As we explain in the following, this can be understood in the frame of the quasi-static Doppler effect presented in Sec. II. Note that whereas in the static Doppler effect regime the frequency difference between two peaks in the scattered wave power spectrum is determined by the scatterer oscillation velocity amplitude, in the quasi-static Doppler effect regime it is determined by the scatterer oscillation frequency. B. Quasistatic Doppler effect FIG. 5. Power spectra of the wave scattered by the vibrating plate as a function of the dimensionless vibration amplitude 2kA 0.39 a, 1.17 b, and 2.34 c k is the wave number of the incident wave of high frequency f. Frequencies were shifted by f 225 khz. For the calculation of k /c we used c 345 m/s for air at ambient temperature T 0 25 1 C. In this section we present experimental results concerning the quasi-static Doppler effect. We show that in our current experimental configuration, which is slightly different than in Ref. 12, the Doppler effect overcomes bulk nonlinearities in the generation of the sidebands at f nf of the backscattered wave. Although the prediction of the phase modulation theory concerning the amplitude of the first sideband is often used in the weak regime 2kA 1 in various applications, 13 18 the phase modulation theory has never been experimentally verified, especially in the strong regime 2kA 1. Three pressure power spectra of the backscattered wave computed for f 225 khz, F 14 Hz, and three different values of the dimensionless amplitude 2kA are displayed in Fig. 5. For the calculation of k /c we used c 345 m/s for air at ambient temperature T 0 25 1 C.) These spectra are composed of pairs of symmetric peaks at f nf surrounding the central peak at f. Comparison of spectra 5a and b shows that the number of sidebands emerging from the background noise around the central peak at f increases with 2kA, exceeding the frequency span for 2kA 1. A first cancellation of the central peak occurs at 2kA 2.3, as shown in Fig. 5 c, where the central peak has decreased by approximately 30 db with respect to its initial value. Figure 6 displays the variations of the normalized amplitudes p n p n /p 0 of the side-bands measured in the power spectrum of the backscattered wave as a function of 2kA, in both the weak a, 2kA 1, and strong b,c, 2kA 1, regimes. Normalization is done using the amplitude p 0 of the wave backscattered on the motionless piston called hereafter the reference wave. Comparison with the predictions of phase modulation theory, i.e., with the absolute values of the Bessel functions, shows excellent agreement, except for high order sidebands when their amplitude is close to the background noise this is for p n 65 db, i.e., p n 6 10 4 ). For 2kA small, the amplitude of the sidebands at frequency f nf (n 0 5) increases like (ka) n /n!. For higher values of 2kA, the amplitude of the component at frequency f of the scattered wave decreases and we observe that it almost vanishes for 2kA 2.3 see Figs. 5 c and 6 b. It then increases when 2kA is increased further above 2.3, and then decreases and vanishes again for 2kA 5.4. J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer 511
Another way to check that the Doppler effect is dominant versus bulk acoustic nonlinearities is to vary the vibration frequency, always keeping Y 1, and to show that the sideband amplitudes p n scale like A, independently of F. This is indeed the case in our experimental configuration. Thus, the Doppler effect is the dominant mechanism in the generation of frequency combinations f nf. We can then study an acoustic nonintrusive estimator of the vibration amplitude A, which is presented in the next section. V. TEST OF A SPECTRAL ESTIMATOR OF THE SCATTERER HARMONIC VIBRATION AMPLITUDE Recently, Huang et al. 15 proposed to use a mathematical property of Bessel functions, z 2 n n 2 J 2 n z, 12 n J 2 n z in order to define a spectral estimator  of the scatterer vibration amplitude A when its motion is harmonic in the quasi-static Doppler regime. In practice, the number of detected sidebands being finite, the estimator has to be defined with a finite number of sidebands 2n 0 :  2k 1 n 0 n n n 2 p 2 0 n 2kA n 0 2 pn 2kA. 13 n n0 We can then test the accuracy and validity domain of this estimator. It is expected to be valid as long as the energy of the undetected sidebands is small compared to the energy of the detected ones. This can be checked by measuring the energy of the backscattered wave E and comparing it to the energy of the reference wave E 0. As a matter of fact, due to the orthogonality of the Bessel functions, the normalized energy of a wave of the form given by Eq. 5 is E E 0 n J 2 n 2kA 1. 14 As long as the undetected energy is small compared to the one of the reference wave, we expect the estimator Ê to be equal to the energy of the reference wave: n 0 Ê n n0 p 2 n 2kA E 0. 15 FIG. 6. Normalized pressure p n p n /p 0 versus 2kA for n 0, 1, 2, 3, 4, and 5. p 0 is the amplitude of the wave backscattered on the motionless piston. a Weak regime in log 10 log 10 scale and b and c strong regime in linear scale. Continuous lines show the absolute values of the Bessel functions of order n 0,...,5. Note that there is no adjustable parameter, and k is fixed by the sound speed c. The same process occurs roughly periodically for the amplitude of each peak f nf, as displayed in Fig. 6 for n 0 5. We note that experiments were performed up to 2kA 30, showing the same behavior. This reflects the fact that the phase modulation process induced by the scatterer vibration does not transfer any energy from the scatterer to the scattered wave. In fact, the energy of the sidebands is pumped from the incident wave only. Figure 7 a displays the average relative error Â/A 1 as a function of 2kA for n 0 5 and 11, where A is determined from the scatterer acceleration. The averages are performed over 22 independent experimental runs for each value of A. This average relative error is compared to the error computed using a theoretical truncated amplitude estimator  T, which is defined with the finite number of detected sidebands: 512 J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer
FIG. 8. Estimated normalized acoustic energy Ê/E 0 versus 2kA for two PSD spans, n 0 5 and 11. In this case the errorbars correspond only to the standard deviation of 22 independent experimental runs. Vertical dashed lines represent the values 2kA n 0. FIG. 7. a Amplitude estimator error, Â/A 1, versus 2kA. The solid line represents the theoretical truncated amplitude estimator error. Vertical dashed lines represent the values 2kA n 0. Error bars correspond to the errors obtained including the standard deviation of 22 independent experimental runs, the errors associated to the amplitude measurements, and the error associated to the value of k 1.5% for c. b Estimated amplitude  versus A. In this case the error bars correspond to the statistical standard deviation of 22 independent experimental runs. In both figures, results for two PSD spans are presented, n 0 5 and 11.  T 2k 1 n 0 n n n 2 J 2 0 n 2kA n 0 2 Jn 2kA. 16 n n0 In the range 2kA n 0, the estimator Â, defined in Eq. 13, is found to have a satisfactory accuracy. The mean value of Â/A 1 does not exceeds 5%. The associated errorbars are important of the order of 20% for 2kA 0.01, but they decrease rapidly to a value below 5% as we increase A. The larger errors are in fact due to the uncertainty of the determination of the amplitude from acceleration measurements at low frequency At both low F and low A, the acceleration is quite small, of the order of 1.6 10 2 m/s 2.) However, the relative errors of both amplitude measurements, i.e., the acceleration and nonintrusive acoustic power spectrum estimator measurements, are of same order of magnitude, as can be seen in Fig. 7 b, where we present  versus A in a log 10 log 10 scalc. In Fig. 7 a we observe that for n 0 5 resp. 11 the estimation error increases above 5% when 2kA 5 resp. 11, in agreement with the error computed using the theoretical truncated amplitude estimator  T /A 1. This departure from 0 is concomitant with the saturation of  observed in Fig. 7 b. These facts are due to the transfer of a significant part of the incident energy to undetected sidebands during scattering. This is clearly observed in Fig. 8, which shows the average value of Ê/E 0 as a function of 2kA. It is compared to the normalized energy computed using a theoretical truncated energy estimator Ê T /E 0, defined as n Ê 0 T E 0 n n0 J 2 n 2kA. 17 Here, we also find that both Ê/E 0 and Ê T /E 0 drop by more than 5% when 2kA n 0. We thus conclude that for the experimental configuration presented here, the acoustic amplitude estimator proposed in Ref. 15 is valid in a large range of amplitudes, namely 2 10 2 2kA n 0, which gives 2 10 3 mm A n 0 /2k 1.2 mm resp. 2.7 mm for n 0 5 resp. n 0 11). The relative error on  is of the same order of magnitude as that on the acceleration measurements. The lower limit of A is given by the signal-to-noise ratio SNR of the acceleration measurements whereas the lower limit of  is given by the PSD SNR, which is of the order of 65 db in our present experimental configuration. Concerning the acoustic nonintrusive estimator Â, for small amplitude measurements (2kA 1) the first sidebands must then satisfy p /p 0 p 1 /p 0 ka 65 db 6 10 4 in order to be resolved. Thus, to get a better resolution on the measured amplitude we have to increase k, i.e., to increase f. 23 On the other hand, to increase the upper limit of the measurable amplitude, one has to in- J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer 513
crease the ratio n 0 /k, either by increasing the number of sidebands taken into account in the analysis or by decreasing k by decreasing f. VI. CONCLUSION We have studied the Doppler effect generated by vibrating scatterers. We have shown that depending on the duration t 0 of the analyzed backscattered wave, we have two possible situations: i if t 0 F 1, the static Doppler effect is observed and the backscattered power spectrum is proportional to the PDF of the scatterer velocity. The frequency shift f is then given by the amplitude of the scatterer velocity. (ii) If t 0 F 1, the so-called quasi-static Doppler effect is observed, and the scattered wave is phase modulated by the normalized vibration displacement 2kx s (t) 2kA sin( t). In this case, the backscattered wave has a frequency spectrum composed of peaks at f nf n integer. Experiments show that the backscattered wave is very well described by Eq. 5, i.e., the amplitudes of the sidebands of order n are given by the Bessel functions of order n. We verified that in the case F f, this agreement holds in both the weak (2kA 1) and strong (2kA 1) regimes. The last part of this work was devoted to the study of a nonintrusive acoustic estimator of the amplitude vibration A. We conclude that for the present experimental configuration, the acoustic amplitude estimator is valid for a large range of amplitudes, namely 2 10 2 2kA n 0, which gives 2 10 3 mm A n 0 /2k O(1 mm). To increase the upper bound one has to take into account a larger number of sidebands or one can chose to decrease the incident sound frequency. For a given SNR, one has to increase the incident frequency to decrease the lower bound. ACKNOWLEDGMENTS RW thanks the Center National d Etudes Spatiales CNES for financial support. This work has been supported by CNES Contract No. 03/11/21/00. 1 A. M. 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Beyer, The parameter B/A, in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock Academic, New York, 1998. 22 The Doppler effect can dominate nonlinear effects for Y 1 in the case of diverging waves because of a less efficient nonlinear interaction in the bulk whereas the Doppler effect remains unchanged. 12 The opposite effect is expected for converging waves. 23 For example, for experiments in water presented in Ref. 12, where the SNR is 80 db, L 20 cm, F 120 Hz, and f 2.25 MHz, using only one sideband, n 0 1, and for a single experimental run, we find that the relative error Â/A 1 is lower than 15% for 7 10 9 m A 5 10 5 m. Smaller errors can be obtained by averaging over several experimental runs as is done the present work. 514 J. Acoust. Soc. Am., Vol. 115, No. 2, February 2004 Wunenburger et al.: Doppler effect by a vibrating scatterer